# Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct Eilenberg-MacLane spectra $HA$, where $A$ is an abelian group. In $S$-modules, this is described in Section IV.2 of this version of EKMM. In symmetric spectra, this is described in Example 1.14 of Stefan Schwede's book project (version 3.0).

Question: Are there models of Eilenberg-MacLane spectra that are fibrant, cofibrant, and (strict) abelian group objects with respect to the addition map $+ \colon HA \times HA \to HA$?

My first candidate was symmetric spectra because there, the construction of $HA$ follows directly from a standard construction of Eilenberg-MacLane spaces $K(A,n)$ as topological abelian groups (or simplicial abelian groups, if working in simplicial sets). In particular, $HA$ is an abelian group object. Moreover, $HA$ is an $\Omega$-spectrum, and those are the fibrant objects in, for instance, the absolute projective stable model structure (Theorem III.4.11 in Schwede's book, or Theorem 3.4.4 in Hovey—Shipley—Smith). However, I'm still missing cofibrancy, and I suspect that a cofibrant replacement would mess up the abelian group object structure.

Another idea would be to use the various stable model structures on symmetric spectra. One could also try in $S$-modules, where every object is fibrant.

For the record, an associative smash product is not crucial to my purposes, though of course it would be nice.

• Hi Martin. I tend to agree that cofibrant replacement in Symmetric Spectra would mess up the group structure, but what about taking cofibrant replacement in the category of monoids? Schwede-Shipley Algebras and Modules give conditions so that bifibrant monoids forget to bifibrant objects, and the condition holds for symmetric spectra (as is shown in Mandell-May-Schwede-Shipley). Alternately, you could try to put a model structure on the category of group objects and do cofibrant replacement there, but I've never done so. Sep 11, 2015 at 12:50
• Hi David, Thanks for the ideas and references. In my case, the monoidal structure is the Cartesian product as opposed to the smash product, which would be something unusual to look at, but still doable. From your references, I also looked at Schwede's Stable homotopical algebra and $\Gamma$-spaces, which might do the trick. I'm dealing with connective spectra, so I can work in $\Gamma$-spaces. Moreover, $HA$ is fibrant in the stable $Q$-model structure. I need to think some more about cofibrancy though. Sep 11, 2015 at 15:56

The following four categories are models for spectra with Eilenberg-MacLane spectra of the desired form.

1. Kan's category of semisimplicial spectra [1]
2. The category $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ of sequential spectra of pointed simplicial sets together with the Kan suspension $\Sigma$
3. The category $\mathbf{Sp}^\mathbb{N}(S^1\wedge -)$ of Bousfield-Friedlander spectra
4. The category $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$ of sequential spectra of pointed weak Hausdorff $k$-spaces $\mathcal{T}$

These four models are connected by Quillen equivalences $$\mathbf{Sp}^\mathbb{N}(S^1\wedge -) \rightleftarrows \mathbf{Sp}^\mathbb{N}(\mathcal{T}) \leftrightarrows \mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma)} \rightleftarrows\{\text{semisimplicial spectra}\}$$ as described by Bousfield and Friedlander in [2]. (Strictly speaking, Bousfield and Friedlander work with the category of sequential spectra of pointed topological spaces instead of $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$, but the stable model structure and the corresponding Quillen equivalences also exist for $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$.)

We elaborate on each model.

1. Semisimplicial spectra: Ken Brown equipped the category of semisimplicial spectra with a model structure in which every object is cofibrant and group objects are fibrant in [3]. Kan's stable Dold-Kan correspondence asserts that the category of abelian group objects of semisimplicial spectra is equivalent to the category of unbounded chain complexes of abelian groups. Considering an abelian group $A$ as an unbounded chain complex concentrated in degree zero yields under this correspondence an abelian group object $HA$ in semisimplicial spectra that models the Eilenberg-MacLane spectrum and that is both cofibrant and fibrant.

2. $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$: There is a right Quillen equivalence $\mathrm{Ps}$ from the category of semisimplicial spectra to the category of sequential spectra $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ with the stable model structure. For any semisimplicial spectrum $X$, the structure maps of $\mathrm{Ps}(X)$ are monomorphism. Thus $\mathrm{Ps}(X)$ is cofibrant. Hence if $X$ is a group object (and therefore in particular fibrant), then $\mathrm{Ps}(X)$ is cofibrant, fibrant and a group object as well, since $\mathrm{Ps}$ is a right Quillen functor. For the semisimplicial spectrum $HA$ above, the sequential spectrum $\mathrm{Ps}(HA)$ is an Eilenberg-MacLane spectrum of the desired form and is explicitly given by the sequence of pointed simplicial sets $$A, \overline{W}A, \overline{W}\overline{W}A, \ldots, \overline{W}^n A,\ldots$$ where $A$ is considered as a constant simplicial abelian group and $\overline{W}$ is "dual" to the right adjoint of Kan's loop group functor.

3. Bousfield-Friedlander spectra: A model for an Eilenberg-MacLane spectrum of the desired form is given by the sequence $$A, BA, BBA, \ldots, B^nA,\ldots$$ where $B$ is the classifying space functor given by the diagonal of the bar construction. The structure map $S^1\wedge B^n A\to B^{n+1}A$ in level $k$ is just the inclusion of the $k$-fold wedge of $(B^n A)_k$ into the $k$-fold product of $(B^n A)_k$. In particular, this model is cofibrant. One way to show that this model is fibrant is to note that it is precisely the Bousfield-Friedlander spectrum construction of the $\Gamma$-space associated to $A$.

4. $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$: The left Quillen functor from any of the two categories of sequential spectra of pointed simplicial sets to $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$ is induced by geometric realization. In particular, it preserves finite products and thus group objects. As a left Quillen functor, it preserves cofibrant objects. It preserves fibrant objects as well. Thus the left Quillen functor applied to any model of an Eilenberg-MacLane spectrum of the desired form in $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ or $\mathbf{Sp}^\mathbb{N}(S^1\wedge -)$ yields a model of the desired form in $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$.

References:

[1] Kan, Semisimplicial spectra

[2] Bousfield and Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets

[3] Brown, Abstract homotopy theory and generalized sheaf cohomology

• Thanks Marc! And welcome to MathOverflow. Your answer is exactly what I was looking for. I just looked at Bousfield—Friedlander's Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, which contains much of the material you describe. Are there other important references that I should look at? Sep 11, 2015 at 22:33